
Let \(\mathbf {L}_{T}^{d-2} = \int _{T} \mathbf {B}_{d-2}(v) \mathbf {B}_{d-2}(v)^{\top } \mathrm {d}v\)  be the \(\binom{d+1}{3} \times \binom{d+1}{3}\)  matrix with entries \(\int _{T}B_{\nu \mu \kappa \delta }^{T,d-2} B_{\nu ^{\prime } \mu ^{\prime } \kappa ^{\prime } \delta ^{\prime }}^{T,d-2} (v)\mathrm {d}v\) .
By the Lemma 15.29 in [1]}, we have
\(\int _{T}B_{\nu \mu \kappa \delta }^{T,d-2}(v) B_{\nu ^{\prime } \mu ^{\prime } \kappa ^{\prime } \delta ^{\prime }}^{T,d-2}(v) \mathrm {d}v & =\frac{{\nu +\nu ^{\prime } \atopwithdelims ()\nu }{\mu +\mu ^{\prime } \atopwithdelims ()\mu }{\kappa +\kappa ^{\prime } \atopwithdelims ()\kappa } {\delta +\delta ^{\prime } \atopwithdelims ()\delta }}{{2d-4 \atopwithdelims ()d-2}} \int _{T}B_{\nu +\nu ^{\prime }, \mu +\mu ^{\prime }, \kappa +\kappa ^{\prime }, \delta +\delta ^{\prime }}^{T,2d-4}(v) \mathrm {d}v \\& = \frac{{\nu +\nu ^{\prime } \atopwithdelims ()\nu }{\mu +\mu ^{\prime } \atopwithdelims ()\mu }{\kappa +\kappa ^{\prime } \atopwithdelims ()\kappa } {\delta +\delta ^{\prime } \atopwithdelims ()\delta }}{{2d-4 \atopwithdelims ()d-2}{2d-1 \atopwithdelims ()3}}V_{T},\) 
